Mathematics > Differential Geometry

Title:The Singular Structure and Regularity of Stationary and Minimizing Varifolds

Abstract: If one considers an integral varifold $I^m\subseteq M$ with bounded mean
curvature, and if $S^k(I)\equiv\{x\in M: \text{ no tangent cone at $x$ is
}k+1\text{-symmetric}\}$ is the standard stratification of the singular set,
then it is well known that $\dim S^k\leq k$. In complete generality nothing
else is known about the singular sets $S^k(I)$. In this paper we prove for a
general integral varifold with bounded mean curvature, in particular a
stationary varifold, that every stratum $S^k(I)$ is $k$-rectifiable. In fact,
we prove for $k$-a.e. point $x\in S^k$ that there exists a unique $k$-plane
$V^k$ such that every tangent cone at $x$ is of the form $V\times C$ for some
cone $C$.
In the case of minimizing hypersurfaces $I^{n-1}\subseteq M^n$ we can go
further. Indeed, we can show that the singular set $S(I)$, which is known to
satisfy $\dim S(I)\leq n-8$, is in fact $n-8$ rectifiable with uniformly finite
$n-8$ measure. An effective version of this allows us to prove that the second
fundamental form $A$ has apriori estimates in $L^7_{weak}$ on $I$, an estimate
which is sharp as $|A|$ is not in $L^7$ for the Simons cone. In fact, we prove
the much stronger estimate that the regularity scale $r_I$ has
$L^7_{weak}$-estimates.
The above results are in fact just applications of a new class of estimates
we prove on the quantitative stratifications $S^k_{\epsilon,r}$ and
$S^k_{\epsilon}\equiv S^k_{\epsilon,0}$. Roughly, $x\in S^k_{\epsilon}\subseteq
I$ if no ball $B_r(x)$ is $\epsilon$-close to being $k+1$-symmetric. We show
that $S^k_\epsilon$ is $k$-rectifiable and satisfies the Minkowski estimate
$Vol(B_r\,S_\epsilon^k)\leq C_\epsilon r^{n-k}$. The proof requires a new
$L^2$-subspace approximation theorem for integral varifolds with bounded mean
curvature, and a $W^{1,p}$-Reifenberg type theorem proved by the authors in
\cite{NaVa+}.